Fomin and Zelevinsky invented cluster algebras in early 2000 in order to find a combinatorial approach to the study of Luzstig and Kashiwara's canonical bases in quantum groups, and to total positivity in semisimple groups. The formalism they developped found many applications beyond the scope of their initial goals. A noticeable example is the fact that Fomin and Zelevinsky discovered a phenomenon of simplification of rational fractions, called « Laurent phenomenon ». In the study of rational sequences defined by recurrence relations (such as the Gale-Robinson sequence or the Somos sequences...), the Laurent phenomenon implies that the sequences under consideration take integer values. A second remarkable example is the proof of Zamolodchikov's periodicity conjecture by B. Keller. Cluster algebras are defined in terms of generators and relations. Contrary to usual presentations, the set of generators and relations is not given a priori. The initial datum is that of an « initial seed » which contains a relatively small subset of the generators (the initial cluster) plus some matrix. That matrix contains all the necessary information in order to construct inductively the whole set of generators, starting from the initial cluster, by means of an operation called « mutation ». The theory of cluster algebras has had fast developments in many directions: Representation theory of quivers, Poisson geometry, integrable systems, Teichmüller spaces, combinatorial polyhedra, algebraic geometry (stability conditions, Calabi-Yau algebras, DT-invariants), Quantum Field Theory... In the present project, we focus on some connections between cluster algebras, algebraic and geometric combinatorics, representation theory, triangulated and monoidal categories and integrable systems. Our objectives are to develop new links between these fields, to study some previously known links, and to obtain new applications. Our project gathers three main themes. A – Riemann surfaces viewed as combinatorial tools: Links with certain triangulated categories (higher generalised cluster categories) and Frobenius categories, or with some combinatorial objects (oriented triangulations and maximal green sequences, multitriangulations, pseudotriangulations). B – Categorification: Monoidal categorification by means of perverse sheaves on Nakajima's graded quiver varieties. Additive categorification, via the definition of a tilting theory in some triangulated orbit categories associated with the combinatorial objects cited above. C – Applications to, and of, cluster algebras: Construction of bases, study of friezes, study of the pentagram map... Our team gathers young mathematicians with various mathematical specialities and with different point-of-views, from geometric combinatorics to triangulated categories, representation theory or integrable systems, who all have a common, deep interest in cluster algebras. In relation to this project, several collaborations, as well as a working seminar, have already begun.

In his address at the ICM in 1994, Maxim Kontsevich stated his Homological Mirror Symmetry Conjecture. This conjecture relates two categories, one from symplectic geometry (the Fukaya category), the other from algebraic geometry (the category of coherent sheaves), via an equivalence of suitably defined derived categories. The conjecture remains wide open to this day. Recently, strong links have been uncovered between homological mirror symmetry and the representation theory of certain classes of associative algebras, called "gentle algebras". This relationship opens new paths to attack long-standing problems in both worlds, allowing the application of well-understood representation theory to the study of Fukaya categories (such as in the search for good generators of the category), and that of geometric tools to study the homological properties of associative algebras (such as the search for numerical derived invariants). In addition to their links with homological mirror symmetry, the class of gentle algebras enjoys a deep relationship with the world of combinatorics and polyhedral geometry. The homological and representation-theoretic properties of these algebras naturally lead to the study of combinatorial objects, such as words on graphs, and geometric ones, such as polyhedra and fans. These objects are closely related to those appearing in the very active field of cluster algebras. This project aims at exploiting interactions between homological mirror symmetry, representation theory of quivers, combinatorial models and polyhedral geometry, brought to light in part by the theory of cluster algebras. The vast knowledge on cluster algebras acquired in the past twenty years opens new directions of research in each of these fields. In the achievement of its objectives, the project will bring together leading experts in the numerous and varied fields involved. More specifically, the aims of the project are the following: - Understand the Fukaya category of certain surfaces using the representation theory of associative algebras. Apply this understanding, among other things, to obtain derived invariants of algebras, to study the Fukaya category of a surface in the non-homologically smooth case and to obtain categorical representations of braid groups. - Push the boundaries of knowledge on the representation theory of gentle algebras and associative algebras in general. In particular, develop tau-tilting theory for infinite-dimensional gentle algebras and more general classes of infinite-dimensional algebras and describe the derived categories of such algebras using cominatorial models. - Describe and study the combinatorial objects related to gentle algebras and cluster algebras. In particular, study their type cones, scattering diagrams, cluster complexes and g-vector fans, and construct polyhedral realisations in finite or infinite types.

"Obtaining the most common epidemiological models (in particular those on which the R0 is done) is based on a decomposition into population compartments. Here, it is split into four compartments of individuals: ""Suspected of being infected"", ""Exposed"", ""Infected"", ""Retired"" (no longer participating in the spread of the virus). In these compartmental models, the only particularity of an individual is therefore the epidemiological class to which he or she belongs. Physical or social heterogeneities are absent. In this project we will therefore question the social bases of the models in order to reintroduce the social gravity in order to provide new indicators of the Covid-19 epidemic and better predict its evolution."

This project aims to foster interactions among the different parts of the broad French holomorphic dynamics community, focusing on the study of moduli spaces associated to algebraic dynamical systems. By their nature, these objects can be approached from multiple perspectives (algebraic, analytic, or arithmetic), and sophisticated techniques in dynamics, whether real, complex, one-dimensional, or multivariate, can be required. Given the rapid development of this topic in recent years, both in France and internationally, our goal is to be a driving force playing a prominent role in this surge. This ambition is grounded in the diverse expertise of our members. The range of skills indeed spans from non-Archimedean geometry to pluripotential theory, through height theory, as well as a deep understanding of bifurcation phenomena such as parabolic implosion or blenders. This project seeks to significantly enhance the interactions among these experts through numerous meetings and three international conferences. It will also promote extended visits by researchers, which are crucial for the development of our international collaborations, and fund young researchers, doctoral students or postdoctoral fellows.

Obtaining the most common epidemiological models (in particular those on which the R0 is done) is based on a decomposition into population compartments. Here, it is split into four compartments of individuals: "Suspected of being infected", "Exposed", "Infected", "Retired" (no longer participating in the spread of the virus). In these compartmental models, the only particularity of an individual is therefore the epidemiological class to which he or she belongs. Physical or social heterogeneities are absent. In this project we will therefore question the social bases of the models in order to reintroduce the social gravity in order to provide new indicators of the Covid-19 epidemic and better predict its evolution.